To investigate the general properties of unpolarized and linearly polarized WW gluon distributions in coordinate space, we typically distinguish two regions. For small dipole size limit, \(\vert r_\perp \vert \ll 1/Q_{sg}\) , one can easily check that \(xG(x_g, r_\perp )\simeq \frac{C_F S_\perp Q^2_{sg}}{2\alpha _s \pi ^2}\ln \frac{1}{\vert r_\perp \vert \Lambda _{\text{QCD}}}\)  and \(xH(x_g, r_\perp )\simeq \frac{C_F S_\perp Q^2_{sg}}{4\alpha _s \pi ^2}\) . In the small \(r_\perp \)  limit, \(xG(x_g, r_\perp )\)  behaves as \(\ln \frac{1}{\vert r_\perp \vert \Lambda _{\text{QCD}}}\)  and \(xH(x_g, r_\perp )\)  to be a constant at given \(x_g\) , the perturbative behaviors are actually in agreement with the properties at large transverse momentum [1]}. Moreover, it is interesting to note that the two gluon distributions have strong dependence on \(A\)  since \(Q^2_{sg}\sim A^{1/3}\)  and \(S_\perp \sim R^2_A\sim A^{2/3}\) , which indicates, in the small \(r_\perp \)  limit, the two gluon distributions have no nuclear shadowing effect, this feature also holds at large transverse momentum. Furthermore, it should be noted that \(\frac{xH(x_g, r_\perp )}{xG(x_g, r_\perp )}\simeq \frac{1}{2}\ln ^{-1}\frac{1}{\vert r_\perp \vert \Lambda _{\text{QCD}}}\) , the ratio seems to be zero when \(r_\perp \)  is small enough even though \(xH\)  still remains non-zero value. Last but not least, the evolutions of the two gluon distributions toward to small-\(x\)  are completely captured by \(Q^2_{sg}\) . However, for \(\vert r_\perp \vert \gg 1/Q_{sg}\) , but \(\vert r_\perp \vert \le 1/\Lambda _{\text{QCD}}\) , one can check that \(xG(x_g, r_\perp )\simeq \frac{2C_F S_\perp }{\alpha _s \pi ^2}\frac{1}{r^2_\perp }\)  and \(xH(x_g, r_\perp )\simeq \frac{C_F S_\perp }{\alpha _s \pi ^2}\frac{1}{r^2_\perp \ln \frac{1}{\vert r_\perp \vert \Lambda _{\text{QCD}}}}\) , the two gluon distributions are independent on \(x_g\)  and fall off as \(r^2_\perp \)  with increasing \(\vert r_\perp \vert \) , these behaviors are completely different from that in small \(r_\perp \)  limit. Also, the ratio \(\frac{xH(x_g, r_\perp )}{xG(x_g, r_\perp )}\approx 1/2\) , which indicates the linearly polarized WW gluon distribution will tend to its saturated boundaryIn practice, the saturated boundary is sensitive to the treatment of logarithmic term \(\ln \frac{1}{\vert r_\perp \vert \Lambda _{\text{QCD}}}\) . If one absorb \(\frac{1}{2}\)  into the logarithmic term, then the saturated boundary tends to 1. when \(r_\perp \)  is large enough.
